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Math  /  Trigonometry

Questionb) sin⁡2x=2sin⁡xcos⁡x\sin 2 x=2 \sin x \cos x c) tan⁡x=sin⁡xcos⁡x\tan x=\frac{\sin x}{\cos x} d) all of these - The height of the tip of one blade of a wind turbine above the ground, h(t)h(t), can be modelled by h(t)=18cos⁡(πt+π4)+2h(t)=18 \cos \left(\pi t+\frac{\pi}{4}\right)+2 where tt is the time passed in seconds. Whic, time interval describes a period when the bl tip is at least 30 m above the ground? a) 5.24≤t≤7.335.24 \leq t \leq 7.33 (c) 1.37≤t≤21.37 \leq t \leq 2. ) 0.42≤t≤1.080.42 \leq t \leq 1.08 d) 0.08≤t≤10.08 \leq t \leq 1.
Iify cos⁡π5cos⁡π6−sin⁡π5sin⁡π6\cos \frac{\pi}{5} \cos \frac{\pi}{6}-\sin \frac{\pi}{5} \sin \frac{\pi}{6}

Studdy Solution

STEP 1

1. We are given a trigonometric identity problem and a modeling problem involving a cosine function.
2. For the trigonometric identity, we will use known identities to simplify the expression.
3. For the modeling problem, we will solve an inequality to find the time interval.

STEP 2

1. Verify the trigonometric identity.
2. Solve the inequality for the height of the wind turbine blade.

STEP 3

Verify the trigonometric identity for part (b) and (c):
b) The identity sin⁡2x=2sin⁡xcos⁡x\sin 2x = 2 \sin x \cos x is a known double angle identity for sine, so it is true.
c) The identity tan⁡x=sin⁡xcos⁡x\tan x = \frac{\sin x}{\cos x} is the definition of the tangent function, so it is true.

STEP 4

Verify the trigonometric identity for part (d):
d) Since both (b) and (c) are true, the option "all of these" is also true.

STEP 5

Solve the inequality for the height of the wind turbine blade:
Given the equation h(t)=18cos⁡(πt+π4)+2 h(t) = 18 \cos \left(\pi t + \frac{\pi}{4}\right) + 2 , we need to find when h(t)≥30 h(t) \geq 30 .
18cos⁡(πt+π4)+2≥30 18 \cos \left(\pi t + \frac{\pi}{4}\right) + 2 \geq 30
Subtract 2 from both sides:
18cos⁡(πt+π4)≥28 18 \cos \left(\pi t + \frac{\pi}{4}\right) \geq 28
Divide both sides by 18:
cos⁡(πt+π4)≥2818 \cos \left(\pi t + \frac{\pi}{4}\right) \geq \frac{28}{18}
Simplify the fraction:
cos⁡(πt+π4)≥149 \cos \left(\pi t + \frac{\pi}{4}\right) \geq \frac{14}{9}
Since 149\frac{14}{9} is greater than 1, there is no real solution for cos⁡(πt+π4)≥149\cos \left(\pi t + \frac{\pi}{4}\right) \geq \frac{14}{9} because the cosine function ranges between -1 and 1.

STEP 6

Since the inequality has no real solution, we conclude that none of the given time intervals satisfy the condition for the blade tip to be at least 30 meters above the ground.
The correct answer for the time interval is none of the provided options.

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